13.472J/1.128J/2.158J/16.940J
COMPUTATIONAL GEOMETRY
Lecture 23
Dr. W. Cho
Prof. N. M. Patrikalakis
c
Copyright 2003
Massachusetts Institute of Technology
Contents
23 F.E. and B.E. Meshing Algorithms
23.1 General . . . . . . . . . . . . . . . . . . . . . . . . . .
23.1.1 Finite Element Method (FEM) . . . . . . . . .
23.1.2 Mesh . . . . . . . . . . . . . . . . . . . . . . . .
23.1.3 Some Criteria for a Good Meshing . . . . . . .
23.1.4 Finite Element Analysis in a CAD Environment
23.1.5 Background Information . . . . . . . . . . . . .
23.1.6 Mesh Conformity . . . . . . . . . . . . . . . . .
23.1.7 Mesh Refinement . . . . . . . . . . . . . . . . .
23.2 Mesh Generation Methods . . . . . . . . . . . . . . . .
23.2.1 Mapped Mesh Generation [23, 10, 11, 5] . . . .
23.2.2 Topology Decomposition Approach [20, 3] . . .
23.2.3 Geometry Decomposition Approach [6, 17, 13] .
23.2.4 Volume Triangulation – Delaunay Based [7, 15]
23.2.5 Spatial Enumeration Methods [21, 22] . . . . .
Bibliography
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1
Lecture 23
F.E. and B.E. Meshing Algorithms
23.1
General
23.1.1
Finite Element Method (FEM)
FEM solves numerically complex continuous systems for which it is not possible to construct
any analytic solution, see Figure 23.1.
Physical Problem
Law of Physics
Formulation of Equations
PDE
Weighted Residual Methods
Integral Formulation
Transformation of Equations
Unknown Function Approximation
(using finite element mesh and matrix organization)
Algebraic System of Equations
Solution
Numerical Solution
Approximate Solution
Figure 23.1: FEM Procedure
2
23.1.2
Mesh
Mesh is the complex of elements discretizing the simulation domain, eg. triangular or quadrilateral mesh in 2D, tetrahedral or hexahedral mesh in 3D.
Why Use Mesh ?
To construct a discrete version of the original PDE problems.
Types of Mesh
See Figure 23.2.
Figure 23.2: Structured (n = 6) and unstructured (n 6= constant) mesh
• Structured mesh
– The number of elements n surrounding an internal node is constant.
– The connectivity of the grid can be calculated rather than explicitly stored.
→ simpler and less computer memory intensive
– Lack of geometric flexibility
• Unstructured mesh
– The number of elements surrounding an internal node can be arbitrary.
– Greater geometric flexibility
→ crucial when dealing with domains of complex geometry or when the mesh has to
be adapted to complicated features of computational domain or to complicated
features of the solution (eg. boundary layers, internal layers, shocks, etc).
– Expensive in time and memory requirements
23.1.3
Some Criteria for a Good Meshing
• Shape of elements [2, 18]
– meshing should avoid both very sharp and flat angles, see Figure 23.3.
– may cause serious numerical problems in both finite element mesh generation and
analysis.
3
Figure 23.3: Elements with sharp and flat angles
• Number of elements
– should be moderate.
– related to efficiency of finite element analysis.
• Topological consistency (homeomorphism) between the exact input domain and its mesh.
– related to robustness of finite element analysis [8, 9].
• Automation, adaptability, etc.
23.1.4
Finite Element Analysis in a CAD Environment
See Figure 23.4.
In Figure 23.4, “input data preparation” includes the preparations of boundary conditions,
loads, and integration constants ∆t etc.; “post-processing of results” includes scientific visualization, etc.
23.1.5
Background Information
Delaunay Triangulation [4, 19]
Given a set of points Pi , the Voronoi region, Vi is a set of points closer to a site Pi than any
other site Pj (See Figure 23.5). Delaunay triangulation is constructed by connecting those
points whose Voronoi regions have a common edge (2D) or face (3D). Some properties of
Delaunay triangulation are:
• it maximizes the minimum angle (globally).
• circumcircle criterion: every circle passing through the 3 vertices of a triangle does not
contain any other vertices.
• it covers the convex hull of all sites.
Mesh Conversion
Performed if a mesh generator produces only one type of element and another type is required.
• Quadrilaterals (Hexahedra) → Triangles (Tetrahedra) [12]: easy and well-shaped mesh,
see Figure 23.6.
• Triangles (Tetrahedra) → Quadrilaterals (Hexahedra)
4
User
Geometric Model Generation
Mesh Generator
FE Mesh
Input Data Preparation
FE Model
Input Data of the Problem
FE System
FE Analysis
Post−Processing of Results
Design Improvement
Mesh Improvement
Figure 23.4: FE analysis in a CAD environment
Voronoi region
Delaunay triangle
Pi
Pj
Figure 23.5: 2D Delaunay triangulation
5
Figure 23.6: Mesh conversion : quadrilateral → triangles
Figure 23.7: New node insertion
– New node insertion [12]: not well-shaped mesh → flat angle, see Figure 23.7.
– Pairing of adjacent triangles (tetrahedra) [14]: pairing test should be performed to
generate convex quadrilaterals. → pairing ∆ABC and ∆CDA is unsuitable, see
A
B
C
D
Figure 23.8: Pairing of triangles
Figure 23.8.
Mesh Smoothing
Make elements better-shaped by iteratively repositioning an internal node Pi , [12].
• Pi =
Pi +
PN
j=1
N +1
Pj
, where N is the number of elements connected by Pi .
→ Pi converges to the centroid of the polygon formed by its connected neighbors.
• Example: After 5 iterations, Pi converges from (7,4) to (4.5,3), see Figure 23.9.
23.1.6
Mesh Conformity
• If adjacent elements share a common vertex or whole edge or a whole face, see Figure
23.10, we have conforming mesh. Otherwise, mesh is non-conforming.
6
y
y
Pi
x
x
Figure 23.9: Mesh smoothing
(a) Conforming meshes
(b) Non−conforming meshes
Figure 23.10: Conforming and non-conforming mesh
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• In case of non-conforming mesh, multiple constraints need to be applied to the midside
nodes to make the mesh analyzable [1], see Figures 23.11 and 23.12.
N2
a
n
b
N1
Figure 23.11: Constrained node, n
→ (eg) Un =
b
U
a+b N2
+
a
U
a+b N1
UN2
Un
N2
UN 1
n
N1
Figure 23.12: Displacement
23.1.7
Mesh Refinement
Increase mesh density from region to region.
Conforming mesh refinement algorithm [16]
Let τ0 be any conforming initial triangulation having N0 triangles. Then a new triangulation
τ1 is defined as follows:
1. Bisect triangle T by the longest side, ∀T ∈ τ0 , see Figure 23.13.
A
D
C
B
Figure 23.13: Step 1
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2. Find the set S1 of triangles generated in step 1 and such that the midpoint of one of its
sides is a non-conforming node, i.e. node D and the corresponding S1 = {∆ABC}, see
Figure 23.13.
3. For each triangle T ∈ S1 , join its non-conforming node with the opposite vertex, i.e. join
D and C, see Figure 23.14.
D
C
Figure 23.14: Step 3, New conforming triangulation
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23.2
Mesh Generation Methods
23.2.1
Mapped Mesh Generation [23, 10, 11, 5]
This is the earliest method for mesh generation which appeared in 1970’s.
Mapped Element Approach
y
v
x
u
Figure 23.15: Mapped element approach
1. Subdivide the physical domain into simple regions, e.g., quadrilaterals, triangles.
2. Generate mesh in computational space (u, v), see Figure 23.15.
3. Determine nodes in the physical domain using shape functions.
• Example 1: Isoparametric curvilinear mapping of quadrilaterals
→ a mapped mesh generation method for surfaces.
v
v
z
u
u
y
x
Figure 23.16: Isoparametric curvilinear mapping
~ = P8 Ni (u, v)X
~ i,
→X
i=1
~ i = (xi , yi , zi ) : coordinates of each node in the physical domain X
~ = (x, y, z)
where X
and Ni (u, v) is the shape function associated with each node, see Figure 23.16.
10
• Example 2: Isoparametic mapping for 3D objects, see Figure 23.17.
z
y
w
w
x
v
v
u
u
(a) Division of a region into zones
(b) Division of a zone into cuboids (c) Division of a cuboid into tetrahedra
Figure 23.17: Isoparametric mapping for 3D objects
Decomposition of 3D objects requires
• user interaction,
• skeleton transform.
Conformal Mapping Approach [5]
Conformal mapping deals with 2D simply-connected regions having more than 4 sides, see
Figure 23.18.
• Algorithm
1. Model a 2D simply-connected region to be meshed with an N-gon P in the complex
w-plane.
2. Define an N-gon Q in the complex z-plane.
3. Find the Schwarz-Christoffel transformation G that maps from the upper half uplane onto the interior of Q, and the transformation F from the upper half plane
onto the interior of P .
4. Map the mesh from Q onto P using the composite mapping w = F (G−1 (z)).
• Note: G−1 is not always found.
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u−plane
z=G(u)
w=F(u)
Q
P
w−plane
z−plane
−1
w=F( G (z) )
Figure 23.18: Conformal mapping
23.2.2
Topology Decomposition Approach [20, 3]
Subtract a simplex from an object repeatedly until the object is reduced to a single simplex.
Simplex in 2D is triangle, while simplex in 3D is tetrahedron.
• (2D Cases) Cut off triangles whose vertices are the object vertices until only 3 vertices
are left, see Figure 23.19.
Figure 23.19: 2D topology decomposition
• (3D Cases)
– Operator OP1 : constructs a tetrahedron by cutting out a corner vertex which has
exactly 3 adjacent edges, see Figure 23.20.
OP1
vertex to be removed
Figure 23.20: Operator OP1
– Operator OP2 : digs out a tetrahedron from a convex edge, see Figure 23.21.
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edge to be removed
OP2
Figure 23.21: Operator OP2
– Note: In either case, the removed tetrahedron must be a part of the object in its
entirety.
– Algorithm
1. Start with a simply connected region.
2. Apply OP1 to all appropriate vertices.
3. If all the remaining vertices have more than 3 adjacent edges, apply OP2 to dig
out a tetrahedron.
4. Go back to step 2 and continue this process until the object is reduced to a
single tetrahedron.
– Note: The topology decomposition approach generates gross elements and the mesh
must be refined later to satisfy the required mesh density distribution.
23.2.3
Geometry Decomposition Approach [6, 17, 13]
Recursive Subdivision
Figure 23.22: Recursive subdivision
1. Start with a convex object, see Figure 23.22.
2. Insert nodes into the boundary of the object to satisfy mesh density distribution.
3. Divide the object roughly in the middle of its longest axis.
4. Insert nodes into the split line according to the mesh density distribution.
5. Recursively divide the two halves until they become triangles
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Iterative Removal of Elements
Figure 23.23: Iterative removal of elements
1. Start with a simply connected region, see Figure 23.23.
2. Insert nodes into the boundary of the object to satisfy mesh density distribution.
3. Remove all vertex angles of the polygon which are less than 90◦ by forming triangular
elements.
4. Start at some vertex with an angle of less than 180◦ but above 90◦ , and form two triangles
with the adjacent points to the vertex.
5. Repeat step 3, step 4 until the last three points form a triangular element.
Iterative Removal of a Boundary Layer [13]
Iteratively remove a boundary layer at a time followed by triangulation, see Figure 23.24.
Figure 23.24: Iterative removal of a boundary layer
• Advancing front methods or paving technique
• Note: Triangulation near boundary is especially well-shaped.
23.2.4
Volume Triangulation – Delaunay Based [7, 15]
2D Case
See page 4.
14
3D Case
1. Insert nodes on each cross-sections, see Figure 23.25.
Figure 23.25: Inserted nodes on cross sections
2. Perform Delaunay triangulation, see Figure 23.26.
Figure 23.26: Delaunay triangulation of points in Figure 23.25
23.2.5
Spatial Enumeration Methods [21, 22]
Based on quadtree (octree) decomposition in 2D (3D), respectively.
A Quadtree Based 2D Meshing Algorithm
• Subdivide the object interior into quadrants whose size satisfy the mesh density distribution.
• Neighboring quadrants may differ by at most one level of subdivision.
• The quadrants on the boundary may have cut corners.
• Finally, each quadrant is broken up into triangles such that the resulting mesh is conforming.
• Note: Excellent interior elements but in general, not well-shaped boundary elements.
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